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Questions and Answers
What is the expansion method used for in determinant evaluation?
In first-order determinants, what happens if you expand along row 1 of a 3x3 matrix?
What does expanding along row 2 of a 3x3 matrix result in for first-order determinants?
How many minors are involved in second-order determinants?
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What is the result when expanding along the last two columns in second-order determinants?
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What happens to the determinant when expanding along the first two columns in second-order determinants?
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How is the determinant calculated when expanding by minors along row 1 for a 3x3 matrix?
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What is the determinant obtained by expanding by minors along row 2 for a 2x2 matrix?
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What does the formula D(M) = ∑(-1)^(i+j) * M_ij * minors of M represent?
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When expanding by minors along row 1 for a 3x3 matrix, what is the correct expression for D(A)?
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For what kind of matrices can the expansion method be generalized according to the text?
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What happens when expanding by minors along row 2 for a 2x2 matrix?
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Study Notes
Determinant Evaluation Up to 3x3 Order
The determinant of a matrix is a scalar value used to describe its orientation and volume properties. It can be computed using several methods, including the expansion by minors along rows and columns. In this article, we will discuss the determinant evaluation up to 3x3 order, with a focus on the expansion method.
Expansion Method
First Order Determinants
In the first order case, there is only one minor and it is equal to the element itself. Thus, the determinant evaluates directly from the elements of the row or column chosen for expansion. For example, if we consider the matrix A = |a b c| and expand by minors along row 1, the determinant D(A) is calculated as follows:
D(A) = a * a - b * b + c * c
This results in a determinant of a^2 - b^2 + c^2. If we choose to expand along row 2 instead, we get:
D(A) = -a * b + b * c - c * a
This also results in the same determinant of a^2 - b^2 + c^2.
Second Order Determinants
In the second order case, we have two minors: one formed by selecting the first two columns and the second formed by selecting the last two columns. When expanding by minors along row 1, we get:
D(A) = (a * d - b * c) - (b * e - c * d)
D(A) = a * d - b * e - (b * e - c * d)
D(A) = a * d - b * e - b * e + c * d
D(A) = (a - b) * d + c * e
Similarly, when expanding by minors along row 2, we get:
D(A) = (b * f - c * e) - (a * f - b * e)
D(A) = b * f - c * e - a * f + b * e
D(A) = (b - a) * f + e * e
D(A) = (b - a) * f + e^2
Third Order Determinants
In the third order case, we have three minors formed by selecting the first three columns, the middle three columns, and the last three columns. When expanding by minors along row 1, we get:
D(A) = (a * g - b * f) - (b * h - c * g) + (b * h - c * i)
D(A) = a * g - b * f - b * h + c * g + b * h - c * i
D(A) = (a - b) * g - b * f - b * h + (b - c) * g + b * h - (b - c) * i
D(A) = (a - b) * g - b * f - b * h + (b - c) * g - b * h + (b - c) * i
D(A) = (a - b) * g - b * f + b * i - (b - c) * h
Generalization
The expansion method can be generalized to any order of the determinant. The determinant of a square matrix M of order n can be calculated using the following formula:
D(M) = ∑(-1)^(i+j) * M_ij * minors of M
Here, the sum is taken over all possible choices of i and j, where i and j are the row and column indices, respectively, and M_ij is the element in the i-th row and j-th column of the matrix M. The minors of M are the determinants of the submatrices obtained by deleting i-th row and j-th column from the matrix M.
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Description
Explore the methods for evaluating determinants of matrices up to 3x3 order, with a focus on the expansion method by minors along rows and columns. Learn how to calculate determinants for 1x1, 2x2, and 3x3 matrices using specific examples.
